Chapter 2 Diagnostic Test

Transcription

1 Chapter Diagnostic Test STUDENT BOOK PAGES Calculate each unknown side length. Round to two decimal places, if necessary. a) b). Solve each equation. Round to one decimal place, if necessary. a) x x x = 69 c) + = b) 4 3 (x + ) 3(5 x) = 6 d) 85 = 6 + x 3. Determine the equation of the line that a) passes through (5, 3) and (8, 6) b) has a slope of and passes through (4, 7) c) is perpendicular to y = 3x and passes through (3, ) d) is parallel to 4x + y = 7 and passes through (, 4) 4. Determine the point of intersection for each pair of lines. a) y = 3x + 5 b) 3y x = y = x 5 x + 4y = 7 5. Calculate the radius of the semicircle. 6. Which of these statements is not true? a) A rectangle is a special type of parallelogram. b) A square is a special type of rhombus. c) A rhombus is a special type of square. d) A rhombus is a special type of kite. Chapter Diagnostic Test

2 Chapter Diagnostic Test Answers. a).5 cm b) 6.3 m. a) or b) 6.7 c) 3 d) 8.3 or a) y = 3x 8 b) y = x + 9 c) y = 3 x d) y = x 6 4. a) ( 8, 9) b) (.7, 3.) 5. about 7 cm 6. c) If students have difficulty with the questions in the Diagnostic Test, you may need to review the following topics: applying the Pythagorean theorem to determine side lengths of a right triangle determining the equation of a line given two points on the line determining the equation of a line given the y-intercept and the slope determining the point of intersection for a linear system Principles of Mathematics 0: Chapter Diagnostic Test Answers

3 Lesson. Extra Practice STUDENT BOOK PAGES Determine the coordinates of the midpoint of each line segment.. A midpoint of a line segment is (3.5, ), and one endpoint is (, 7). a) Describe a strategy you could use to determine the coordinates of the other endpoint. b) Apply your strategy to determine these coordinates. 3. a) The points (, 4) and ( 3, ) are the endpoints of a diameter of a circle. Determine the coordinates of the centre of the circle. b) Another diameter of the same circle has endpoint (, 8). Determine the coordinates of the other endpoint. 4. Three of the vertices of rhombus ABCD are A(5, 6), B(, 5), and C(3, 0). a) What property of rhombuses can you use to determine the coordinates of the fourth vertex, D? b) Determine the coordinates of D. 5. Determine the equation of the perpendicular bisector of the line segment with each pair of endpoints. a) (3, ) and (5, 5) b) ( 3, ) and (5, ) c) (3, 3) and (7, 3) 6. The municipal councils of two cities agree that a new airport will not be closer to one city than to the other. The coordinates of the cities on a map are A(3, 7) and B(47, 5). Describe all the possible locations for the airport. 7. Quadrilateral PQRS has vertices at P(, 7), Q(6, 8), R(7, ), and S(3, ). a) Determine the point of intersection for the diagonals of this quadrilateral. b) Determine the midpoint of each diagonal. c) Is PQRS a parallelogram? Explain how you know. Lesson. Extra Practice 3

4 Lesson. Extra Practice Answers. midpoint of AB: (3, 4) midpoint of CD: (3.5,.5) midpoint of EF: (, 0.5). a) Use the midpoint formula to write an equation for each coordinate: x + x y + y = 3.5 and =. Substitute x = and y = 7 into these equations, and solve for x and y. b) (6, ) 3. a) (4, 3) b) (9, ) 5. a) y = x + 5 b) y = x c) x = 5 6. The airport can be located at any point on the line with equation y = 3x a) (4.5, 3.5) b) midpoint of PR: (4, 4) midpoint of QS: (4.5, 3.5) c) No. The diagonals do not meet at a common midpoint. 4. a) Opposites sides of a rhombus are parallel. b) (0, ) 4 Principles of Mathematics 0: Lesson. Extra Practice Answers

5 Lesson. Extra Practice STUDENT BOOK PAGES Determine the length of each line segment.. Calculate the distance between every possible pair of points in the diagram. 3. The coordinates of four possible sites for a landfill are given below. Which site is farthest from a town located at (4, 8)? a) (0, 5) b) (7, ) c) (6, 3) d) (8, 4) 4. Suppose that you are given the coordinates of the endpoints of a line segment. Describe the most efficient strategy you could use to calculate both the slope and the length of the line segment. 5. The walls of a room in an art gallery need to be repainted. The coordinates of the corners of the room, in metres, are (3, 8), (7, 7), (7, ), (, ), and (, 5). The room is 5 m high, and L of paint covers 35 m. How much paint will be needed, to the nearest tenth of a litre? 6. An optical fibre trunk runs in a straight line through points (5, 0) and (45, 0) on a grid. What is the minimum length of optical fibre that is needed to connect the trunk to buildings at (30, 35) and (0, 0), if unit on the grid represents m? Round your answer to the nearest tenth. Lesson. Extra Practice 5

6 Lesson. Extra Practice Answers. PQ: 5.8 units; RS: about 4. units TU: about 4. units. AB: about 4.5 units AC: about 8.5 units AD: about 3.6 units BC: about 6. units BD: about 6. units CD: about 7. units 3. d) 4. Answers may vary, e.g., determine x x and y y. For the slope, divide the second value by the first value. For the length, square the values, add them, and take the square root L m 6 Principles of Mathematics 0: Lesson. Extra Practice Answers

7 Lesson.3 Extra Practice STUDENT BOOK PAGES This table of values gives points that lie on the same circle. x 7 5 y ± ±5 ±7 ± a) Copy and complete the table. b) Sketch the circle. c) Write the equation of the circle.. For each equation of a circle: i) Determine the diameter. ii) Determine the x- and y-intercepts. a) x + y = 65 b) x + y = A circle is centred at (0, 0) and passes through point ( 3, 7). Write the equation of the circle. 6. A satellite transfers from a near-earth orbit, with equation x + y = 5 500, in kilometres, to another orbit that is 35 km higher. How many kilometres longer, to two decimal places, is the second orbit than the first? 7. A circle is centred at (0, 0) and passes through point (6, 7). Determine the other endpoint of the diameter through (6, 7). Explain your strategy. 8. Points (a, 7) and (4, b) are on the circle with equation x + y = 50. Determine the possible values of a and b. Round to one decimal place, if necessary. 4. Which of these points is not on the circle with equation x + y = 5? a) (4, ) b) (, 9) c) (0, 5) d) (9, ) 5. a) Determine the radius of a circle that passes through i) (0, 6) ii) (5, ) b) Write the equation of each circle in part a). c) State the coordinates of two other points on each circle. Lesson.3 Extra Practice 7

8 Lesson.3 Extra Practice Answers. a) x b) y ± ±5 ±7 ±7 ±5 ± 3. x + y = a) 5. a) i) 6 units ii) 3 units b) i) x + y = 36 ii) x + y = 69 c) Answers may vary, e.g., i) (0, 6), ( 6, 0) ii) (3, 0), ( 3, 0) km c) x + y = 50. a) i) 50 units ii) x-intercepts: (5, 0), ( 5, 0) y-intercepts: (0, 5), (0, 5) b) i) 0.8 units ii) x-intercepts: (0.4, 0), ( 0.4, 0) y-intercepts: (0, 0.4), (0, 0.4) 7. ( 6, 7); the diameter has midpoint (0, 0), so the coordinates of the other endpoint are the opposites of the coordinates of the given endpoint. 8. a = ±, b = ±5.8 8 Principles of Mathematics 0: Lesson.3 Extra Practice Answers

9 Chapter Mid-Chapter Review Extra Practice STUDENT BOOK PAGES The midpoint of line segment AB is at M( 5,.5). If endpoint A is at (, 3), determine the coordinates of endpoint B.. Rectangle EFGH has vertices at E(, 5), F(9, 7), and G(8, ). Determine the coordinates of the fourth vertex, H. 3. A guide rope is laid out in three stages along the ascent of a vertical rock face. The rope is attached to the rock face at points (0, 0), (5, 3), (, ), and (3, 7), in that order. Determine the total length of the rope, to the nearest tenth of a metre. 6. Write the equation of a circle, centred at (0, 0), that models each situation. a) a G-training device for astronauts, with a radius of 3 m b) a wheel rim with a diameter of 6 in 7. A design for a kitchen tile uses circles and squares, as shown. If the large square has side lengths of 5 mm, what is the equation of the small circle, assuming that its centre is at (0, 0)? Round to the nearest millimetre. 4. The Big Pumpkin, a restaurant and pie shop, is a short distance from a major highway. The highway passes, in a straight line, through ( 4, ) and (8, 4) on a map. The Big Pumpkin is located at (, 3). What is the shortest distance from The Big Pumpkin to the highway, to the nearest tenth of a kilometre, if unit on the map represents km? 5. Two points, A(0.5,.5) and B(3.5, 3.5), lie on a line. Point C( 0.5, 5.5) lies off to one side of the line. a) Use only the distance formula to show that BC is not perpendicular to the line through A and B. b) Determine the distance from C to the line through A and B. Do not calculate a point of intersection. Chapter Mid-Chapter Review Extra Practice 9

10 Chapter Mid-Chapter Review Extra Practice Answers. ( 9, 8). (0, 9) m 6. a) x + y = 69 b) x + y = x + y = 953, or x + y = km 5. a) Answers may vary, e.g., AC = 0 = 3. 6 or about 3. units and BC = 5 = or about 4.5 units so BC is not the minimum distance from C to AB. Therefore, BC is not perpendicular to AB. b) Answers may vary, e.g., the slope of AB = 3 and the slope of AC = 3, which means that AC is perpendicular to AB. The distance from C to AB is the length of AC, which is about 3. units. 0 Principles of Mathematics 0: Chapter Mid-Chapter Review Extra Practice Answers

11 Lesson.4 Extra Practice STUDENT BOOK PAGES Quadrilateral ABCD has two sides that measure 5 units and two sides that measure.5 units. What types of quadrilateral could ABCD be? For each type, make a possible sketch of ABCD on a grid.. P( 3, ), Q(, 3), and R(, 7) are the vertices of a triangle. a) Show that +PQR is an isosceles triangle. b) Is +PQR also a right triangle? Justify your answer. 3. a) Describe a strategy you could use to decide whether a quadrilateral is a rhombus, if you are given the coordinates of its vertices. b) Use your strategy to show that polygon JKLM, with vertices at J( 5, 3), K(3, ), L(5, 7), and M( 3, 5), is a rhombus. 4. In quadrilateral EFGH, EF = 7.5 cm and FG = 5.5 cm. The slopes of EF, FG, and GH are 4, 4, and 4, respectively. a) What must be true about the slope and length of EH if EFGH is a parallelogram? b) Could the length of GH equal 7.5 cm if EFGH is an isosceles trapezoid? Explain. 5. Two vertices of +ABC, an isosceles right triangle, are located at A(, ) and B(5, ). Determine all the possible locations of C if a) AB is a side of+abc that is not the hypotenuse b) AB is the hypotenuse 6. A rhombus is a special type of kite and also a special type of parallelogram. How would you apply this description of a rhombus if you were using the coordinates of the vertices of a quadrilateral to determine the type of quadrilateral? Include examples in your explanation. 7. The corners of a new park have the coordinates S( 5, ), T(5, 4), U(, 5), and V( 5, ) on a map. What shape is the park? Include your calculations in your answer. Lesson.4 Extra Practice

12 Lesson.4 Extra Practice Answers. Answers may vary, e.g., parallelogram rectangle kite. a) PQ = PQ = 6 = 5. 0 units, PR = PR = 6 = 5. 0 units b) No. Answers may vary, e.g., m PQ =, 5 m PR = 5, and m QR = ; none of the slopes are negative reciprocals. 3. a) Determine all four side lengths, and check whether they are equal. b) JK = KL = LM = JM = 7 = 8. 5units 4. a) m EH = 4, EH = 5.5 cm b) No. Since EF and GH are the parallel sides in the isosceles trapezoid, they cannot be the same length. 5. a) (5, 4), (, ), (8, ), or (, 5) b) (, ) or (5, ) 6. Answers may vary, e.g., I would use the distance formula to determine the lengths of all the sides. If two pairs of adjacent sides are congruent, then the quadrilateral is a kite; e.g., A( 3, 3), B(, 5), C(7, 3), D(, ). I would use the slope formula to determine the slopes of all the sides. If the slopes of opposite sides are equal, then the quadrilateral is a parallelogram; e.g., E(, ), F(4, 5), G(8, 5), H(5, ). If the quadrilateral is both a kite and a parallelogram, then it is a rhombus; e.g., J(, ), K(4, 6), L(7, ), M(4, 4). 7. ST = 5 5 =. 8 units, TU = 7 = 4. units, UV = 3 5 = 6.7 units, SV = 3 units; since the side lengths are all different, the park is not a parallelogram, kite, or isosceles trapezoid. m ST =, m TU =, m UV =, m SV is 4 undefined; since the park has one parallel pair of sides, it is a (non-isosceles) trapezoid. Principles of Mathematics 0: Lesson.4 Extra Practice Answers

13 Lesson.5 Extra Practice STUDENT BOOK PAGES a) Rectangle ABCD has vertices at A( 3, ), B(0, ), C(5, 3), and D(, 6). Show that the diagonals are the same length. b) Give an example of a quadrilateral that is not a rectangle but has diagonals that are the same length.. a) Show that the diagonals of quadrilateral EFGH bisect each other at right angles. b) Make a conjecture about the type of quadrilateral in part a). Use analytic geometry to explain why your conjecture is either true or false. 3. Kite PQRS has vertices at P( 3, 3), Q(, 3), R(, 5), and S( 6, ). Show that the midsegments of this kite form a rectangle. 4. Show that the diagonals of the kite in question 3 are perpendicular, and that diagonal PR bisects diagonal QS. 5. a) Show that points A(5, 5), B( 5, 5), C(, 7), and D( 7, ) lie on the circle with equation x + y = 50. b) Show that AB is a diameter of the circle. c) Show that AB is the perpendicular bisector of chord CD. 6. A trapezoid has vertices at J( 4, 4), K(0, 7), L(8, 3), and M(8, ). a) Show that the line segment joining the midpoints of JK and LM bisects the line segment joining the midpoints of JM and KL. b) Show that the two line segments described in part a) are perpendicular. c) Based on your answer for part b), make a conjecture about trapezoid JKLM. Use analytic geometry to determine whether your conjecture is true. 7. +XYZ has vertices at X(3, 5), Y(3, ), and Z( 3, 4). Verify that the area of the parallelogram formed by joining the midpoints of the sides of +XYZ and the vertex X is half the area of +XYZ. Lesson.5 Extra Practice 3

14 Lesson.5 Extra Practice Answers. a) AC = 7 = 8. 5 units, BD = 7 = 8.5 units b) Answers may vary, e.g., a quadrilateral with vertices at (0, ), (4, 0), (0, 6), and ( 4, 0): both diagonals have a length of 8 units, but the quadrilateral is not a rectangle.. a) meg = 3 and m FH = 3, so EG and FH are perpendicular; M EG = (.5,.5) = M FH, so EG and FH bisect each other. b) Answers may vary, e.g., conjecture: EFGH is a rhombus. EF = FG = GH = EH = 5 units, so my conjecture is correct. 3. Answers may vary, e.g., M PQ = ( 0.5, 3), M QR = (.5, ), M RS = (.5, 3), M PS = ( 4.5, ); the slopes of the four midsegments are,,, and, so adjacent midsegments are perpendicular. Therefore, the midsegments form a rectangle. 4. Answers may vary, e.g., m PR = and m QS =, so the diagonals are perpendicular. The equation of the line through PR is y = x 3, and M QS = (, ) lies on this line (check by substitution). Therefore, PR intersects QS at its midpoint. 5. a) For A(5, 5), 5 + ( 5) = 50; for B( 5, 5), ( 5) + 5 = 50; for C(, 7), + 7 = 50; for D( 7, ), ( 7) + ( ) = 50; the points lie on the circle. b) M AB = (0, 0), which is the centre of the circle, so AB is a diameter. c) Answers may vary, e.g., m AB = and m CD =, so AB and CD are perpendicular. The equation of the line through AB is y = x. M CD = ( 3, 3), so M CD lies on this diagonal (check by substitution) and therefore AB intersects CD at its midpoint, i.e., AB is the perpendicular bisector of CD. 6. a) Answers may vary, e.g., M JK = (, 5.5) and M LM = (8, 0.5), so the equation of the line through M JK and M LM is 9 y = x +. MKL = (4, 5) and M JM = (, ), so the midpoint of M KL and 9 M JM is (3, 3), which lies on y = x + (check by substitution). Therefore, M JK M LM intersects M KL M JM at its midpoint, i.e., M JK M LM bisects M KL M JM. b) The slope of M M JK LM = and the slope of M KL M JM =, so M JK M LM and M KL M JM are perpendicular. c) Answers may vary, e.g., conjecture: trapezoid JKLM is isosceles. Since m = = and JK = 5 = LM, my KL m JM conjecture is correct. 7. Answers may vary, e.g., the parallelogram has vertices at X(3, 5), M XY = (3,.5), M YZ = (0, 3), and M XZ = (0, 0.5). Let XM XY be the base of the parallelogram, so the base length is 3.5 units, the height is 3 units, and the area of the parallelogram is 0.5 square units. Let XY be the base of +XYZ, so the base length is 7 units, the height is 6 units, and the area is square units. Therefore, the area of the parallelogram is half the area of +XYZ. 4 Principles of Mathematics 0: Lesson.5 Extra Practice Answers

15 Lesson.7 Extra Practice STUDENT BOOK PAGES 5. In +ABC, the altitude from vertex B meets AC at point D. a) Determine the equation of the line that contains AC. b) Determine the equation of the line that contains BD. c) Determine the coordinates of point D. d) Determine the area of +ABC, to the nearest whole unit.. A fountain is going to be constructed in a park, an equal distance from each of the three entrances to the park. The entrances are located at P(, 9), Q(, 6), and R(3, ). Where should the fountain be constructed? 3. +XYZ has vertices at X(6, 5), Y(4, ), and Z(, ). a) Determine the equation of the perpendicular bisector of XY. b) Points P, Q, and R are on the perpendicular bisector from part a). Determine the coordinates of these points that make the distances PX, QY, and RZ as short as possible. c) Point S is also on the perpendicular bisector from part a). Determine the coordinates of S that make the distances SX, SY, and SZ equal. 4. A circle with radius r is centred at (0, 0). The circle has a horizontal chord, AB, with endpoints A( h, k) and B(h, k) and a vertical diameter, CD, with endpoints C(0, r) and D(0, r). a) Draw a diagram of the circle, showing AB, CD, and their point of intersection, E. Include expressions for the lengths of the line segments in your diagram. b) Use properties of the intersecting chords of a circle to verify that the coordinates of A and B satisfy the equation of the circle. 5. Classify the triangle that is formed by the lines y = x + 3, x + y = 6, and 3y x 4 = 0 a) by its angles b) by its sides 6. The arch of a bridge, which forms an arc of a circle, is modelled on a grid. The supports are located at ( 5, 0) and (5, 0), and the highest part of the arch is located at (0, 9). What is the radius of the arch, if each unit on the grid represents m? Lesson.7 Extra Practice 5

16 Lesson.7 Extra Practice Answers. a) y = 3 x + b) y = x c) (0.5, 0.5) d) 5 square units 5. a) right triangle b) isosceles 6. 7 m. (6, 6) 3. a) y = 0.5x b) P(5, 3), Q(5, 3), R(0., 5. 4) c) S(, 5) 4. a) b) ( AE)( BE) = ( CE)( DE) ( h)( h) = ( r k)( r + k) h h + k = r = r k 6 Principles of Mathematics 0: Lesson.7 Extra Practice Answers

17 Chapter Review Extra Practice STUDENT BOOK PAGES 5. Show that the line defined by x + y = 0 is the perpendicular bisector of line segment AB, with endpoints A(6, ) and B(, 6).. Rhombus CDEF has vertices at C(, ), D(, 6), and E(5, 9). a) Determine the coordinates of vertex F. b) Determine the perimeter of the rhombus. 3. A sonar ping travels outward from a submarine in a circular wave at 550 m/s. Assuming that the submarine is located at (0, 0), write an equation for the sonar wave after 5 s. 4. +ABC has vertices as shown. Determine whether this triangle is isosceles, equilateral, or scalene. 5. a) A quadrilateral is formed by the lines y = x + 4, y x + 3 = 0, y x = 5, and a fourth line. Explain why this quadrilateral cannot be a parallelogram. b) If the quadrilateral in part a) is a trapezoid, what must be true about the fourth line? 6. a) Show that +PQR, with vertices at P(, 4), Q(, 9), and R(5, ), is isosceles. b) Suppose that M and N are the midpoints of PQ and PR. Explain, without calculations, why +MNP is also isosceles. 7. +XYZ has vertices as shown. Determine the orthocentre of this triangle. 8. +DEF has vertices at D(0, 0), E( 5, k), and F( 5, k), where k is a positive number. The circumcentre of +DEF is at (0, 0). a) Determine the equation of the circle that passes through D, E, and F. b) Determine the value of k. c) Determine whether +DEF is isosceles, equilateral, or scalene. Explain your answer. Chapter Review Extra Practice 7

18 Chapter Review Extra Practice Answers. Answers may vary, e.g., the slope of the line defined by x + y = 0 is, and m AB =, so AB is perpendicular to the line. Since M AB = (4, ) lies on the line (check by substitution), the line is the perpendicular bisector of AB.. a) F(, 5) b) 0 units 3. x + y = scalene 5. Answers may vary, e.g., a) No two of the three given lines are parallel, so it is impossible for both pairs of opposite sides of the quadrilateral to be parallel. b) The slope of the fourth line must be,, or. 6. a) PQ = PR = 5 units, so +PQR is isosceles. b) Answers may vary, e.g., MP =, PQ =, PR = NP, so +MNP is isosceles. 7. (5, 3) 8. a) x + y = 00 b) 5 3 or about 8.66 c) equilateral; DE = 0 3 = 7.3 or about 7.3 units, EF = 0 3 = 7.3 or about 7.3 units, DF = 0 3 = 7.3 = or about 7.3 units 8 Principles of Mathematics 0: Chapter Review Extra Practice Answers